Conversations Conversations among among Inference Relations Inference Relations Itala M. Loffredo D’Ottaviano itala@cle.unicamp.br Centre for Logic, Epistemology and the History of Science University of Campinas Algebraic Semantics for Uncertainty and Vagueness IRSES PROJECT Salerno, May 2011 1
The method of studying inter- relations between logical systems by the analysis of translations between them was originally introduced by Kolmogorov, in 1925. Kolmogorov, A.N. (1977) On the principle of excluded middle (1925). In: HEIJENOORT, J. (Ed.) From Frege to Cambridge: Harvard University Press, p. 414-437. G ö del: a source book in mathematical logic 1879-1931. 2
respect to intuitionistic logic. Some of them were developed mainly consistency of classical logic with relative the show to order in (1933) and Gentzen (1933). The by Kolmogorov (1925), Glivenko (1929), involving classical logic, intuitionistic ‘translations’ known first logic and modal logic were presented Lewis and Langford (1932), G ö del 3
In spite of Kolmogorov, Glivenko, dealing with inter-relations between the systems studied by them, they are not interested in the meaning of the concept of translation between logics. Since then, interpretations between logics have been used to different purposes. G ö del and Gentzen 4
these historical papers and this is the first paper in which a general definition for the concept of translation between logical systems is introduced. connections between classical, intuitionistic and minimal logic. In: SCHMIDT, H. et alii . (Ed.) Contributions to mathematical logic . Amsterdam: North-Holland, p. 215- 229. PRAWITZ AND MALMN Ä S Prawitz and Malmn ä s (1968) survey Prawitz, D., Malmn ä s, P.E. (1968) A survey of some 5
are the first works with a general systematic study on translations between logics. Both study inter-relations between propositional calculi in terms of translations. W ó jcicki (1988) and Epstein (1990) 6
The ”Campinas CLE - Group“ Definition of Translation Carnielli, W.A., D’Ottaviano, I.M.L., Alves, E.H. (1997) Translation between logics: a manifesto. Logique et Analyse , v. 40, p. 67-81 . 7
LOGICS AND TRANSLATIONS for translation. the essential feature of a logical to single out what seems to be in fact translation between logics, in order of concept the definition Da Silva, D’Ottaviano and Sette (1999), general a propose general, in inter-relations between logic systems ex plicitly interested in the study of 8
consequence relations. between preserving maps as defined are logics translations Logics and operator, consequence formulas of a language) and a that in general a logic deals with constituted by a set (ignoring the fact are characterized as pairs 9
Definition: A logic A is a pair < A , C >, where the set A is the domain of A and C is a consequence operator in A , that is, C: P (A) → P (A) is a function that satisfies, for X, Y ⊆ A : (i) X ⊆ C (X ) (ii) X ⊆ Y , then C ( X ) ⊆ C ( Y ) (iii) C ( C ( X )) ⊆ C (X ) 10
Definition: A translation from a logic A into a logic B is a map t : A → B such that, for any X ⊆ A t ( C A ( X )) ⊆ C B ( t ( X )). 11
associated syntactic consequence , respectively, then If A and B are formal languages, with relations ⊢ and ⊢ C C B A t is a translation if, and only if, for Γ∪ { α } ⊆ Form ( A ): Γ ⊢ α implies t ( Γ ) ⊢ t ( α ). C C B A 12
Pure and Applied Mathematics, v. 203) between New York: Marcel Dekker, p. 435-448. (Lectures Notes in MONTENEGRO, C.H. (Eds.) Models, algebras and proofs . X., CAICEDO, In: logics. Translations An initial treatment of a theory of da Silva, J.J., D’Ottaviano, I.M.L., Sette, A.M. (1999) and Sette (1999). presented by da Silva, D’Ottaviano is logics between translations 13
An important subclass of translations, the conservative translations, was investigated by Feitosa and D’Ottaviano. 14
Definition: Let A and B be logics. A conservative translation from A into B is a function t : A → B such that, for every set X ∪ { x } ⊆ A : x ∈ C A ( X ) if, and only if, t ( x ) ∈ C B ( t ( X )) 15
Feitosa, H.A., D’Ottaviano, I.M.L. (2001) Conservative translations. Annals of Pure and Applied Logic , Amsterdam, v. 108, p. 205-227. D’Ottaviano, I.M.L., Feitosa, H.A. (1999) Conservative Internacional de Filosofia, v XXII, n.2, p. 117-132. Feitosa, H.A. (1997) conservativas (Conservative translations). Doctorate Thesis. Campinas: Institute of Philosophy and the Human Sciences, State University of Campinas. translations and model-theoretic translations – Revista Tradu çõ es
Note that, in terms of consequence conservative translation when, for relations, t : Form ( L 1 ) → Form ( L 2 ) is a every Γ∪ { α } ⊆ Form ( L 1 ): Γ ⊢ α if, and only if, t ( Γ ) ⊢ t ( α ). C C 2 1 17
translation were imposed. and the forgetful map from modal conservative of notion stricter Such cases would be ruled out if the logics into classical logic. from intuitionistic into classical logic Our translations, such as the identity map seem to be intuitive examples of accommodates certain maps that translation of notion 18
In this sense, the more abstract notion and general concept of translation that we have assumed is a genuine advance in the scope of relating logic systems, based upon which further unfoldings can be devised. 19
Translations in the sense of Prawitz translations in our sense. particular cases of our conservative translations. Epstein`s translations are instances of and Malmn ä s do not coincide with Translation in W ó jcicki s sense are our conservative translations. 20
But i is not a conservative translation: it Example 1 suffices to observe that while The identity function i : IPC → CPC , both logics considered in the connectives ¬ , ∧ , ∨ , → , is a translation from IPC into CPC: for every Γ⊆ Form ( L ), C IPC ( Γ ) ⊆ C CPC ( Γ ). p ∨¬ p ∉ C IPC ( ∅ ) I ( p ∨¬ p ) = ( p ∨¬ p ) ∈ C CPC ( ∅ ). 21
However i : CPC → IPC is not a translation 22
Kolmogorov s, Glivenko s and Gentzen s interpretations are conservative translations from classical into intuitionistic logic. 23
are not translations in our sense, even in the propositional level. modal interpretation of intuitionistic logic. Anthology of Universal Logic : from Paul Hertz to Dov Gabbay. Springer Basel: Studies in Universal Logic . Both G ö del s (1933) interpretations D`Ottaviano, I.M.L., Feitosa, H.A. (2011) On G ö del`s 24
Some General Results on Conservative Translations The next results are relevant to the study of general properties of logic systems from the point of view of translations between them. 25
consistent, then L 1 consistent. Proposition : If t : L 1 → L 2 is a literal translation relatively to ¬ and L 2 is ¬ is ¬ – – 26
logic systems, the next result corresponds to the compactness of the systems. When A 1 and A 2 are strongly complete Theorem : If A 1 and A 2 are logics with finitary consequence operators, t : A 1 → A 2 is a conservative translation if, and only if, for every finite A ∪ { x } ⊆ A 1 , x ∈ C 1 ( A ) is equivalent to t (x) ∈ C 2 ( t ( A )). 27
conservative if, and only if, for every The following theorem supplies a necessary and sufficient condition for a translation between deductive systems being conservative. Theorem* : A translation t : A 1 → A 2 is A ⊆ A 1 , t -1 ( C 2 ( t ( A ))) ⊆ C 1 ( A ). 28
from a non-vacuum system into a Proposition: T here is no translation vacuum system. 29
Theorem : If there is a recursive and conservative translation from a logic system L 1 into a decidable logic As an easy consequence, there is no recursive conservative translation from first-order logic into CPC . system L 2 , then L 1 is decidable. 30
and there is a surjective and conservative translation L 2 . Conservative translations preserve non-triviality. Proposition : If L 1 is a logic system with an axiomatic Λ t : L 1 → L 2 , then t ( Λ ) is an axiomatic for 31
Preservation of Deduction Meta- Theorems Theorem: Conservative translations preserve the Deduction Theorem.
An Important Algebraic Result By dealing with the Lindenbaum algebraic structures associated to logics, Feitosa and D Ottaviano conservative translations. obtained a useful method to define 33
Given a logic A , consider the equivalence relation on A and the quotient map x ∼ y = def C ( x ) = C ( y ) Q : A → A ~ 34
A 2/ A 1/~ 1 injective. Moreover, if such t* exists, then it is . ~ translation t *: and only if, there is a conservative 2 be the logics co-induced by and A 2/~ 2 A 1/~ 1 and let Theorem : Let A 1 and A 2 be logics, with the domain of A 2 being denumerable; A 1 , Q 1 and A 2 , Q 2 respectively. Then there is a conservative translation t : A 1 → A 2 if, → 35
Based on the previous results, Feitosa and D Ottaviano, dealing with syntactic results, algebraic semantics and matrix semantics, have introduced conservative translations involving: Families of Conservative Translations
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